Question

Question Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8 , y=3x−4 ?(1 point) Responses (−4,−16) and (−3,−13) left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis (4,8) and (3,5) left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis (4,8) left parenthesis 4 comma 8 right parenthesis (3,5)

Answer 1

To solve the linear-quadratic system using the substitution method, we can substitute the expression for \( y \) from the linear equation into the quadratic equation.

Given the equations:

1. \( y = x^2 - 4x + 8 \)
2. \( y = 3x - 4 \)

We can set them equal to each other:

\[ x^2 - 4x + 8 = 3x - 4 \]

Now, rearranging the equation to set it to zero:

\[ x^2 - 4x - 3x + 8 + 4 = 0 \]

This simplifies to:

\[ x^2 - 7x + 12 = 0 \]

Next, we factor the quadratic:

\[ (x - 3)(x - 4) = 0 \]

So the solutions for \( x \) are:

\[ x = 3 \quad \text{and} \quad x = 4 \]

Now, we substitute these \( x \) values back into the linear equation \( y = 3x - 4 \) to find the corresponding \( y \) values.

1. If \( x = 3 \):

\[ y = 3(3) - 4 = 9 - 4 = 5 \]

So one solution is \( (3, 5) \).

2. If \( x = 4 \):

\[ y = 3(4) - 4 = 12 - 4 = 8 \]

So the other solution is \( (4, 8) \).

Thus, the solutions to the system of equations are:

\[ (3, 5) \quad \text{and} \quad (4, 8) \]

The correct response is:

\[ (4, 8) \text{ and } (3, 5) \]